I've been reading Alexander Berzin's articles on his site recently and found them very informative. I think his articles, with plenty of examples, helped me get a better grasp of some Abhidharma concepts, which hitherto, were mostly just words for me. Maybe others find it easy to learn in the "lemma, definition" etc way, but I find that way of pedagogy very difficult even in subjects like mathematics; examples or at least metaphors help me a lot.

Anyway, in this article and a few others, he talks about how the Western definition of "mindfulness" actually involves a few different mental factors in Abhidharma. Within Abhidharma itself, smrti is not really defined the same way by Vasubandhu or Asanga, and in practice, all the Tibetan schools actually follow Asanga's definition in shamatha practice.

I shall lay out my understanding of smrti below, please check for any flaws:

First of all, we need to establish some fundamental terms in Abhidharma; in particular for talking about smrti, we need to be clear about the difference between vijnana/rnam shes (primary consciousness) and caitisika dharma/sems byung (mental factors). Caitasika in Sanskrit means derived from citta, semss byung in Tibetan means the same thing -- arisen from sems. Therefore, the name itself implies that it arises from the main mind and is secondary, something which "mental factor" in English doesn't quite bring across.

So the six vijnanas (putting aside the 2 Chittamatra vijnanas for now) are all primary and the 51 caitasika dharmas accompany them. For example, when our eye-consciousness cognizes a piece of gold, there can be negative caitasika dharmas accompanying this vijnana such as raga (attachment) or positive such as smrti if we are meditating on a gold statue of the Buddha. When our taste-consciousness cognizes a salty taste, we might have raga, if we like salt, or pratigha, if we do not. In a rough metaphor, I guess we could say that the caitasika dharmas are kinda like lenses, while the vijnanas are like our eyes? E.g. we can look at an ant with yellow-coloured lenses, binoculars, a microscope, but the ant is still an ant.

Smrti/dran pa can be compared to a "mental glue" and is accompanied by samprajanya and samjna. For Asanga, smrti is something that only occurs with positive vijnanas that we are familiar with, and it basically ties our vijnana to the object and prevents our mind from wandering on some train of thought. So if we are practicing shamatha on a Buddha statue, we have our eye-vijnana with the physical statue as the object as well as our mind-consciousness with the cognition of the eye-vijnana as the object; if it is a visualized statue, then just the latter -- but in practice, in both cases we are applying smrti, samjna, and samprajanya to the latter.

If we are beginners, then our smrti is weak, and will get distracted by thoughts. Samprajnya is the caitasika dharma which "guards" our smrti, as we progress through the nine stages of shamatha, our samprajnya develops and helps redirect our mind-vijnana to the object whenever it wanders more and more quickly, and our smrti also gets stronger, so it doesn't leave the object. Samjna helps us distinguish the object of our shamatha from e.g the feeling of our legs on the meditation seat, the wind etc. Apramada is what makes us care about returning to the object of our shamatha if our samprajnya finds that our mind-vijnana is distracted. So as a rough metaphor, let's say we are using a telescope to look at the moon, we want it to be fixed on the moon, and we have lenses that bring the moon into focus if it loses it, a homing lens on the moon and so on.

So basically all these other caitasika dharmas accompanying smrti are completely normal and even desireable in shamatha. I was under the impression that they were "thoughts" and a sign that my shamatha was still very shallow, but from what I understand now, they are not problems at all. They are not additional objects of mind-vijnana, which is the real problem e.g. a song lyric appearing in my head, and then I go after the song, think about where I last heard it, the life of the singer, and so on. Subtle thoughts about the object are also a problem such as e.g. thinking about where the Buddha statue came from etc and should not be confused with the caitasika dharmas.

The common Western understanding of "mindfulness" in daily life is closer to the Vaibasika/Theravadin idea as opposed to Asanga's definition, and involves applying smrti to each moment of cognition. But aside from that, the same caitasika dharmas also arise and are fine.

Is this correct? This is for shamatha with an object.

For shamatha without an object, I am unsure. Berzin said this

On mind (mental activity) itself, unaimed at objects of cognition as if they existed on their own.

The latter method of focusing for attaining shamatha is used in mahamudra (phyag-chen, great seal) and dzogchen (rdzogs-chen, great completeness) meditations. There are at least four major manners of meditating:

In the Karma Kagyu tradition of mahamudra, we focus first on commonsense objects constructed from the sensibilia of each of the senses (the sight of an orange, the smell of an orange, the taste of an orange, and so on) and then on a visualized object. When we gain a stable level of concentration, we then focus with it on the mind itself, but without being aimed at the mind as an object. We do this by settling down into the mind’s natural state of bliss (bde-ba), clarity (gsal-ba), and bareness (stong-pa).
In the Sakya tradition of mahamudra, we stare at a visual object and then focus on just the clarity aspect (gsal-ba) of the cognition, which is the aspect that is giving rise to the cognitive appearance.
In the Gelug/Kagyu tradition of mahamudra, we focus on the superficial nature (kun-rdzob, conventional nature) of mind as the mental activity of merely giving rise to cognitive appearances and cognitively engaging with them (gsal-rig-tsam, mere clarity and awareness).
In the Nyingma tradition of dzogchen, we settle down into the natural state in between thoughts.

The final sentence about "natural state in between thoughts" does not sound right to me, but I'm not sure how appropriate it would be to talk about Dzogchen instructions here. Anyway, in terms of relating it to Abhidharma, I guess shamatha without an object would actually be applying smrti to the alayavijnana? This is just a guess, since in many texts, a common warning is that "objectless" meditation very often is just meditating on the alaya.

If we are beginners, then our smrti is weak, and will get distracted by thoughts. Samprajnya is the caitasika dharma which "guards" our smrti, as we progress through the nine stages of shamatha, our samprajnya develops and helps redirect our mind-vijnana to the object whenever it wanders more and more quickly, and our smrti also gets stronger, so it doesn't leave the object. Samjna helps us distinguish the object of our shamatha from e.g the feeling of our legs on the meditation seat, the wind etc. Apramada is what makes us care about returning to the object of our shamatha if our samprajnya finds that our mind-vijnana is distracted. So as a rough metaphor, let's say we are using a telescope to look at the moon, we want it to be fixed on the moon, and we have lenses that bring the moon into focus if it loses it, a homing lens on the moon and so on.

So basically all these other caitasika dharmas accompanying smrti are completely normal and even desireable in shamatha. I was under the impression that they were "thoughts" and a sign that my shamatha was still very shallow, but from what I understand now, they are not problems at all. They are not additional objects of mind-vijnana, which is the real problem e.g. a song lyric appearing in my head, and then I go after the song, think about where I last heard it, the life of the singer, and so on. Subtle thoughts about the object are also a problem such as e.g. thinking about where the Buddha statue came from etc and should not be confused with the caitasika dharmas.

The common Western understanding of "mindfulness" in daily life is closer to the Vaibasika/Theravadin idea as opposed to Asanga's definition, and involves applying smrti to each moment of cognition. But aside from that, the same caitasika dharmas also arise and are fine.

The different sūtras in accord with the emptiness
taught by the Sugata are definitive in meaning;
One can understand that all of those Dharmas in
which a sentient being, individual, or person are taught are provisional in meaning.

From what I've read and understood I think your understanding is basically correct in the context of shamatha with an object.

Many of the caitasika dharmas that you speak of can be found among the 5 faults and 8 antidotes with the important addition of Vigilance / shé shyin which is very similar to Sampajañña. (I don't know for sure, but I imagine vigilance is a subset of dran-pa/smrti and of Sampajañña if it's not a listed mental factor? Many of the terms seem to overlap when translated )

If you're fully going for the 9 stages approach then apparently the faults of excitement (Auddhatya / göpa ) and dullness are only done away with at the 8th stage at which point they are very subtle excitation and dullness. Until then, these 8 antidotes including dran pa/smrti must be applied to the 5 faults.

"All these terms - mindfulness, vigilance, forgetfulness, and all the rest have very precise meaning in the context of samatha. These meanings are very different from the general meaning or those implied in other contexts.'

'As the [practice proceeds, one's chief concern should be with mindfulness, focusing on the object of mediation. However one part of one's awareness should always be devoted to vigilance. If you err in this technique giving too much emphasis to vigilance, what should be an antidote will become an obstruction, and mindfulness will deteriorate'

'Mindfulness is focused on the chief object, be it the breath of the Buddha image. In contrast, vigilance is focused on the meditation its self, the quality of the meditative awareness. It is intermittently checking up to see whether that meditative awareness is too tight or too loose'

'It is essential to have an understanding of the definitions of laxity and excitement, and yet mere intellectual understanding is not enough. In addition, one needs to ascertain each mental factor as it arises during meditation. Even Beyond that, it is important to acquire the more subtle capacity to recognize laxity and excitement when they are on the verge of arising.
It is through the gradual cultivation of vigilance that one is able to recognize laxity and excitement, first as they arise, and second when they are on the verge of arising.' - Samatha Meditation by Gen Lamrimpa

J. Buddha: “And what monks, is the faculty of sati ? Here, monks, the noble disciple has sati , he is endowed with perfect sat i and introspection, he is one who remembers, who recollects what was done and said long before.” ( Sa ṃ yutta Nik ā ya V 197 - 8)

K. Nāgasena: “ Sati , when it arises, calls to mind wholesome and unwholesome tendencies, with faults and faultless, inferior and refined , dark and pure, together with their counterparts... sati , when it arises, follows the courses of beneficial and unbeneficial tendencies: these tendencies are beneficial, these unbeneficial; these tendencies are helpful, these unhelpful. Thus, one who practic es yoga rejects unbeneficial tendencies and cultivates beneficial tendencies.” ( Milindapañha 37 - 38)

L. Buddhaghosa: “By means of [ sati ] they [i.e., other mental processes] remember, or it itself remembers, or it is simply just remembering, thus it is sati . It s characteristic is not floating; its property is not losing; its manifestation is guarding or the state of being face to face with an object; its basis is strong noting or the close applications of mindfulness of the body and so on. It should be seen as l ike a post due to its state of being set in the object, and as like a gatekeeper because it guards the gate of the eye and so on.” ( The Path of Purification , XIV, 141)

M. Buddha: “Herein a monk should constantly review his own mind thus: ‘Does any excitation concerning these five cords of sensual pleasure ever arise in me on any occasion?’ If, on reviewing his mind, the monk understands: ‘Excitation concerning these five cords of sensual pleasure does arise in me on certain occasions’ then he understands: ‘Desire and lust for the five cords of sensual pleasure are not abandoned in me.’ In this way he has introspection of that. But if, on reviewing his mind, the monk 4 understands: ‘No excitation concerning these five cords of sensual pleasure arises in me on any occasion,’ then he understands: ‘Desire and lust for the five cords of sensual pleasure are abandoned in me.’ In this way he has introspection of that.” ( Majjhima Nik ā ya 122 15)

N. Asaṅga: “Mindfulness and introspection are taught, for the first prevents the attention from straying from the meditative object, while the second recognizes that the attention is straying.”

O. Asaṅga: “What is mindfulness? The non - forgetfulness of the mind with respect to a familiar object, having the function of non - distraction.” [ Ab hidharmasamuccaya , Pradhan, ed., p. 6.6.]

P. Śāntideva: “In brief, this alone is the definition of introspection: the repeated examination of the state of one’s body and mind.